Recently, I’m rethinking a very inspiring dialogue in:

especially, the reply below that I posted one year ago

Since I know @DoubleBass read some DCP literature, @valerio88 nicely answered @littlemacro and my naive questions last year, posts of @dgroll were exactly what I thought, and I’m always deeply indebted to @jpfeifer , hope you and other dynare users may be interested in these questions Welcome all comments

Allow me to write my questions in next posts. Please be patient, thx!

I’m particularly eager to know what I should write in the red rectangle, if I reformulate the problem in a Rotemberg setting, instead of Calvo as shown in the paper.

Now I modify the pricing block under the Rotemberg.
The intermediate-good producer wants to max the profit in the home market: \displaystyle\max_{\{P_{HHt}(\omega)\}} \mathbb{E}_t \sum_{t=0}^{\infty} \Lambda_{0,t} \left\{ \left[ P_{HHt}(\omega) - \mathcal{MC}_{t} \right] Q_{HHt}(\omega) -\frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2 P_{HHt}Q_{HHt} \right\}
, and in the foreign market: \displaystyle\max_{\{P^*_{HFt}(\omega)\}} \mathbb{E}_t \sum_{t=0}^{\infty} \Lambda_{0,t} \left\{ \left[ \mathcal{E}_tP^*_{HFt}(\omega) - \mathcal{MC}_{t} \right] Q^*_{HFt}(\omega) -\frac{\xi_P}{2}\left[\frac{P^*_{HFt}(\omega)}{ P^*_{HF,t-1}(\omega)}-\bar{\Pi}^*_{HF}\right]^2 \mathcal{E}_tP^*_{HFt}Q^*_{HFt} \right\}.
After we impose symmetric equilibrium, all (\omega) will be removed. We can obtain profits per period: prof_{HHt}=\left(P_{HHt}- \mathcal{MC}_t\right) Q_{HHt}-\frac{\xi_P}{2}\left(\Pi_{HHt}-\bar{\Pi}_{HH}\right)^2 P_{HHt}Q_{HHt}, prof^*_{HFt}=\left(\mathcal{E}_tP^*_{HFt}- \mathcal{MC}_t\right) Q^*_{HFt}-\frac{\xi_P}{2}\left(\Pi^*_{HFt}-\bar{\Pi}^*_{HF}\right)^2 P^*_{HFt}Q^*_{HFt},

On the other hand, for the Home Bundler, suppose the final goods are \mathcal{F}_t. Authors said: \mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{Q_{HHt}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{Q_{FHt}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}.
Note, according to the arrows, it’s Q_{FHt} rather than Q^*_{HFt}. Also, \mathcal{F}_t=C_t+X_t+G_t

BUT after multiplying \mathcal{MC}_t on both sides, \mathcal{MC}_tY_t = P_tX_t+W_tL_t=\mathcal{MC}_t\left(Q_{HHt} + Q^*_{HFt}\right) + \mathcal{MC}_t \left\{ \frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2 Q_{HHt} + \frac{\xi_P}{2}\left[\frac{P^*_{HFt}(\omega)}{ P^*_{HF,t-1}(\omega)}-\bar{\Pi}^*_{HF}\right]^2 Q^*_{HFt} \right\}

The first line of the equalities will cancel the relevant terms in prof_{HHt} and prof^*_{HFt}. HOWEVER the second line is \mathcal{MC}_t*Rotemberg, not the pure Rotemberg, hence has no chance to cancel terms in prof_{HHt} and prof^*_{HFt}.

It implies the aggregate country budget constraint will be wrong!

Furthermore, do I need change \mathcal{F}_t to: \mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{\tilde{Q}_{HHt}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{\tilde{Q}_{FHt}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}
in which, because of the Rotemberg friction / deadweight loss, \tilde{Q}_{HHt}= \displaystyle\frac{Q_{HHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2}
and \tilde{Q}_{FHt}= \displaystyle\frac{Q_{FHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{FHt}(\omega)}{ P_{FH,t-1}(\omega)}-\bar{\Pi}_{FH}\right]^2}
??

Are you doing higher order solutions or trend inflation? If not, most of the differences in the budget constraint do not matter, because the quadratic costs will drop out.

Because Rotemberg pricing involves actual resource costs, you need to decide in which units these costs are paid. Here, you have some leeway. It’s up to you to have them paid in different or identical units in different sectors. You just have to make sure to include appropriate relative prices when adding them up.

In fact, I believe I have already found out the solution
Thank you for replying me , and you prevent my embarrassment from answering my own questions (s.t. all posts in one topic were solely from me )
Please be tolerant I asked the same questions to another professor, waiting for his reply. In any circumstances, I’ll come back and make out my thought in few days.